Thursday, January 9, 2014

The Residue Theorem and Complex Integrals

Recall that we saw the “residue theorem” (due to Cauchy) Let f(z) be analytic on and inside a closed contour C (see diagram) except for a finite number of isolated singularities at z = a1, a2…..etc., which are enclosed by C.

òCf(z)dz
=2 piånk = 1Res f (ak)

We
now want to elaborate this a bit more by reference to the diagram shown. In
this case we consider the function f(z) is analytic inside and ON the simple
closed curve C except at a finite number of specified points: a, b, c,
etc.at which there exist residues:a- 1,b- 1 ,c- 1,
etc.

In
which case we can write:

òCf(z)dz
=2 pi[a- 1+b- 1+c- 1+ …………………….]

That
is, 2 pitimes the sum of the residues at all the
singularities enclosed by C. To ensure this, one would respectively construct
circles C1, C2, C3 etc. as I have done with respective centers at a, b, c etc.
If we take care to do this properly then we can write:

òCf(z)dz =òC1f(z)dz+ òC2f(z)dz+ òC3f(z)dz+ ..........

Where:

òC1f(z)dz=2 pia- 1

òC2f(z)dz=2 pib- 1

òC3f(z)dz=2 pic- 1

So
that:

òCf(z)dz
=2 pi[a- 1+b- 1+c- 1+ ..] =

2
pi(sum
of residues)

Example
1:

Evaluate the integral:òCcot (z)dz

f(z)
= cot (z)

For
which: òCf(z)dz=2 pic- 1

Re-write:
f(z) = cot (z) = 1/ tan z

For
which singularities occur at tan z = 0

Or:
o, +p, + 2p,+3petc.

Then
Res f(z) =1/ sec2 z ÷ z = +n p=1/ (1/ cos2 z)

=
cos2 z÷ z = +n p=cos2 (np)

And
:cos2 (np)= 1at z =(2n + 1) p)/ 2

Therefore:
c- 1=1, and

òCcot (z)dz=2 pi(1) = 2 pi

Example
2:

Evaluate
the integral:

òCexp (z)dz/(z – 1) (z + 3)2

Where
C is given by÷ z ÷=3/2

Solution:

Take
the residue at the simple pole (z = 1) such that:

limz ® 1[ (z – 1)exp (z)/ ( z -1) (z
+ 3)2 ] =

exp(1)/ 16 = e/ 16

The
residue at the 2nd order pole (z = -3) is:

limz ® -3d/ dz [(z + 3)2 exp (z)/ ( z -1) (z + 3)2 ] =

limz ® -3[ (z – 1)exp (z)- exp(z) / (z – 1 )2
]

= - 5 exp (-3) / 16

The
integral is therefore:

òCexp (z)dz/(z – 1) (z + 3)2 =2 pia- 1=2 pi(e/
16)

(We
do not add the 2nd residue because it lies beyond the circle ÷ z ÷=3/2)

Problems for Math
Mavens:

1)
Evaluate the integral:òC(z + 1) dz / (2z +i)

2)
Consider Example (2) and obtain the integral if we have ÷ z ÷=10instead of

About Me

Specialized in space physics and solar physics, developed first astronomy curriculum for Caribbean secondary schools, has written thirteen books - the most recent:Fundamentals of Solar Physics. Also: Modern Physics: Notes, Problems and Solutions;:'Beyond Atheism, Beyond God', Astronomy & Astrophysics: Notes, Problems and Solutions', 'Physics Notes for Advanced Level&#39, Mathematical Excursions in Brane Space, Selected Analyses in Solar Flare Plasma Dynamics; and 'A History of Caribbean Secondary School Astronomy'. It details the background to my development and implementation of the first ever astronomy curriculum for secondary schools in the Caribbean.